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Is infinity an odd or even number? (2011)

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Re: Is infinity an odd or even number? (2011)

#261

Post by someweirdperson »

bombolo wrote: post #90
yamtaddle wrote: post #69

Earlier quoted context omitted.

Yes, but you'll see "decimal" more in the wild, and that's what people mean by it. "You write it with a decimal point", and they do usually mean to include the irrationals. So, yes, real numbers, but the reasoning behind their usage is "you write it with a decimal point". I'd bet more people understand "decimal number" used in that sense, than understand "real number".

i've never seen an irrational number written in digits in my whole life. Have you????? I've seen them expressed as letters or formule

> i've never seen an irrational number written in digits in my whole life. Have you?????

I'm currently reading one. Looking good so far. I'll let you know after I finish.

Re: Is infinity an odd or even number? (2011)

#262

Post by NegativeK »

tesseract wrote: post #259
NegativeK wrote: post #180

Earlier quoted context omitted.

No math instruction I had ever discussed infinity with any rigor until calculus -- and even then, it was only infinity as a limit. Infinity as a concept was brushed off in the same way that the square root of negative one was brushed off until we were actually taught about it.

On the one hand I get why that is - the calculus notion of infinity is the one that tends to be useful in applied math - on the other hand it's a shame because the set theoretic notion of infinity has more to offer to someone trying to ponder the nature of the infinite. Or put another way, "what's ∞ + 1" basically invites the non-answer "that's not a well-formed question" whereas "what's ω + 1" gives you a whole inte…

I've always been disappointed that number theory, set theory, etc aren't introduced in middle school or high school.

It makes sense, since those are a lot less useful than the subjects that are taught, but something like number theory is incredibly approachable to a middle school student. And it can show students that math can be a lot less about memorization and a lot more about creative thinking w.r.t. proofs.

Re: Is infinity an odd or even number? (2011)

#264

Post by feoren »

dbtc wrote: post #235
tobiasSoftware wrote: post #36

Earlier quoted context omitted.

There are, and it turns out that this is a significant mathematical concept. The integers between 0 and infinity are defined as "countably infinite". Other infinities are considered countably infinite, or the "same" infinity, if and only if you can arrange it in a list such that each item in the list pairs to an integer in our 0 to infinity list. So the set of even numbers is countably infinite because for every i th…

"Countably infinite" makes zero sense to me. Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers. I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem. And pairing one infinite list with another infinite list doesn't make either one any mo…

> Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers.

Exactly correct! This holds true of everything you can generate stepwise, even infinite sets. Cantor proved that you cannot "generate" (stepwise) all Reals between 0 and 1. Any infinite set you can generate stepwise is Countably Infinite.

> I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem.

Exactly correct! And then infinitely later, when you're "done", having generated every Real between 0 and 1, he will then generate a new Real not on your list. Oops! You have not generated all Reals between 0 and 1, even with infinite time.

> And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.

Exactly correct! Any two sets you can pair together (via a bijection) have the exact same cardinality. Neither is more infinite nor countable than the other. Cantor proved you cannot "pair" the Reals with the Natural Numbers.

You and Cantor agree completely. You're very close to understanding why the Reals are bigger.

Re: Is infinity an odd or even number? (2011)

#265

Post by singularity2001 »

logifail wrote: post #236
singularity2001 wrote: post #51

Earlier quoted context omitted.

Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives (including algebraic derivative of step functions without dirac 'density') and yes: natural addition and multiplication. https://en.wikipedia.org/wiki/Hyperreal_number

> Transfinite ordinals also known as hyperreals should really be taught in school as they make many parts of math easier: algebraic definition of derivatives Q: What proportion of children study maths long enough to understand derivatives?

I can only speak for Germany where over 90% reach 10th grade, where derivatives are taught.

Re: Is infinity an odd or even number? (2011)

#266

Post by gus_massa »

C-x_C-f wrote: post #210
gus_massa wrote: post #58

The problem with transfinite is that you lose commutatively. Flowing the standard notation, where the usual infinite in the integer or the real line is "ω = ∞ = 1,2,3,..." ω+1 = ω+1 , i.e. "the next thing after infinity" 1+ω = ω , i.e. "the same infinity as before" 2ω = ω , i.e. "the same infinity as before", so it's even 1+2ω = ω , i.e. "the same infinity as before", so it looks odd, but don't fall in that trap ω2 =…

> After lunch, I went to teach limits to first years students, and with a total straight face I told them that ∞ is not a number. When you apply Alexandroff extension to add the point at infinity to, say, the real numbers, what you're left with is not a set of numbers (i.e. a field) anymore. So it makes sense to say that ∞ is not a number. Moreover, the way ∞ is used in analysis is different from Alexandroff compacti…

It was a long time ago, something like an optional course in Advanced Functional Analysis. It was about the algebras of functions with and without unity, and how to complete the ones without unity using the compactification (i.e. including a ∞) and a few variants.

> two infinities (±∞)

It depends. In the real numbers it depends, but in most cases I agree that it's better to use two. In complex analysis it's much better to have only one infinity. And there are more weird case like the projective plane where you have one infinity in each direction.

> So it makes sense to say that ∞ is not a number.

I agree, it's not longer a field and the operation lose many properties if you try to extend them. So I said "(almost) a number". Anyway, the weird part is that in some cases you can write f(∞) in an advanced math course, but you can never write f(∞) in a fist year math course.

Re: Is infinity an odd or even number? (2011)

#267

Post by thriftwy »

tsoukase wrote: post #252
zw123456 wrote: post #202

The way I would explain it to a 6 year old would be like this: Infinity isn't a number really, it's a concept, like the word many or the word few. If someone says they have many of something, you don't think is that odd or even you just know they have a lot of it. Infinity is kind of like that, it explains the idea of things going on forever, not an exact quantity of things like the number 10 or 11.

My 6 and 7 yo's call infinity the "endless number". Well, at least it is a NaN number :) PS: they seem to _know_ that endless*endless > endless but do not dare to admit it

Infinite is just a fancy word for endless, anyway.

Re: Is infinity an odd or even number? (2011)

#268

Post by feoren »

lisper wrote: post #86

To explain to a six-year-old I would start by telling them that there are many different kinds of infinity, not just one. Some infinities are odd, others are even, and others are neither. It matters whether you are asking "how many" (cardinals) or "in what position" (ordinals). For regular finite numbers, cardinals and ordinals are (more or less) the same, but for infinities they behave differently. Then, if they wan…

> It matters whether you are asking "how many" (cardinals) or "in what position" (ordinals)

But "even" and "odd" are all about whether you can partition something into an equal number of pairs or not. If you're asking "in what position" (ordinals), you've explicitly said you're not in the realm of counting sets of things. I would argue division makes no sense in the realm of ordinals! Everyone is saying the transfinite ordinals alternate even-odd, but those are exactly the numbers where we've stated we're only interested in position, not counting. It's not clear to me why "dividing" an ordinal number into equal pairs makes any sense. (Whereas it makes perfect sense for cardinal numbers.)

Re: Is infinity an odd or even number? (2011)

#269

Post by ForOldHack »

Ahh, the mis-uses of infinity again. Infinity, is both simply because inf+1 = inf, so if infinity is odd, then infinity + 1 is even, which equals infinity which is then odd. Think of sets. Inf and -inf are in both sets. You can prove this with deltas and epsilons, but that is beyond the scope of explaining it to 6 year olds.
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